Bulk Diffusion in Alumina: Solving the Corundum Conundrum

Transcription

1 Bulk Diffusion in Alumina: Solving the Corundum Conundrum Nicholas D.M. Hine 1,2,3 K. Frensch 3, W.M.C Foulkes 1,2, M.W. Finnis 2,3, A. H. Heuer 3,4 1 Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge 2 Condensed Matter Theory Group, Department of Physics, Imperial College London, and 3 Thomas Young Centre, Department of Materials, Imperial College London 4 Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH. Electronic Structure Discussion Group Wednesday 3rd June 2009

2 Outline Introduction 1 Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 2 Self-Consistency 3 Migration Barriers Calculating Effects of Aliovalent Doping 4

3 Outline Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 1 Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 2 Self-Consistency 3 Migration Barriers Calculating Effects of Aliovalent Doping 4

4 Alumina Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina α-al 2 O 3 : Corundum Structure (2:3 coordination 1/3 Al sites vacant). Traditionally thought of as fully ionic: (Al 3+ ) 2 (O 2 ) 3. Large band gap, fully occupied O 2p bands, empty Al 3s/3p bands.

5 Point Defects Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Ceramic crystals formed and annealed at high temperatures always contain nonzero concentrations of intrinsic point defects, even when pure V O, V Al, O i, Al i,... (1) Concentrations of defects depend on annealing conditions, p O2, T. Equilibrium concentrations minimise total Gibbs Free Energy G subject to constraints of charge neutrality and particle number.

6 Point Defects Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Alumina is the prototypical refractory ceramic. Properties as a refractory oxide strongly depend on defects and their diffusion coefficients: Electrical Conductivity Grain Growth Plastic Deformation Oxide film growth Sintering, Creep

7 Point Defects Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Intrinsic defect population in pure alumina can only form charge neutral combinations. In traditional, fully-ionic picture these are: Schottky (vacancies) and anti-schottky (interstitials) 3V O V Al 3 and 3Al i O i 2 (2) Cation Frenkel (Al ions) and Anion Frenkel (O ions) V Al 3 + Al i 3+ and V O 2+ + O i 2 (3) In these combinations, variations in atom chemical potentials due to annealing conditions also cancel, since formation energies contain zero net contribution from µ Al and µ O (NB 2µ Al + 3µ O = µ Al2O 3 )

8 Doping Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Aliovalent dopants (eg Mg and Ti) affect balance of charge: e.g. Replace neutral trivalent Al atom with neutral tetravalent Ti. 3 valence electrons from Ti Al complete 2p shells of nearby O 2. One valence electron would remain in Ti 4s defect state below conduction band edge (6-8 ev above VBM). System can lower free energy by creating a -ve point defect and transferring electron to this. Ti Al is source of electrons & prompts formation of -ve defects.

9 Doping Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Likewise, divalent Mg Al only donates 2 electrons and thus requires an extra electron to come in from elsewhere to fill O 2p shells of neighbours. Mg Al is acceptor of electrons & prompts formation of +ve defects? Remember that the O 2p hole is above top of VBM so filling depends on Fermi energy.

10 Clusters Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Substitutional impurity and intrinsic defect have opposite charge and may bind to form defect clusters, such as (Ti Al : V Al ) 2 and (Mg Al : V O ) 1+ (4) However, even though g f for the reaction is usually -ve, entropy may outweigh it at finite T Clusters dissociate at high temperatures.

11 Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Point Defect Concentration and Migration Law of Mass Action gives concentrations c i of each species in terms of formation energy Gi f per defect: c i = m i e G f i /kt Formation energies also determine diffusion coefficients by Arrhenius Equation D i = Di 0 e E i a /kt where activation energy Ei a = Gi f + Gi b, where Gi b is the migration barrier between adjacent equivalent positions of the defect.

12 The Corundum Conundrum Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Despite widespread industrial usage, defect phenomena in Al 2 O 3 not well-understood or well-controlled. Diffusion experiments are our only experimental window to understanding, but low intrinsic concentrations, lack of good radiotracer for Al, and unknown impurities content of samples limit interpretation of results. Previous Classical Potential and DFT calculations produced formation energies very hard to reconcile with experiment (E a 6eV).

13 The Corundum Conundrum Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Some clear facts emerge from experiments D O is smaller than D Al (or D for cation impurities) by several orders of magnitude Rate-limiting D loop sensitive to balance of aliovalent impurities. Up by 100 with 250ppm Mg; down by with 600ppm Ti. However, given very low intrinsic concentrations 10 10, these are remarkably small changes Could native defect concentrations somehow be buffered against increasing concentrations of dopants? What role do defect clusters play? Can we predict diffusion coefficients ab initio?

14 Formation Energies Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina However Gi f depends on chemical potentials of species transferred to and from reservoirs to create defect i: G f i = G def i α (n α + n α i )µ α + q i µ e G def i : Gibbs energy of supercell with defect i, charge q i. µ e, µ α : chemical potential of electrons & atom species α G f i = G def i G perf i α n α i µ α + q i µ e G perf i : Total energy of supercell of perfect crystal = α nα µ α

15 Formation Energies Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina So, each G f i, and thus the c i s depend on µ e, µ O and µ Al. µ O and µ Al are functions of annealing conditions: Determine via combination of Ideal Gas Eq for µ 1 2 O 2(g) at T, p O 2 (g), and experimental and theoretical values for G f (Al 2 O 3 ) (see work of Finnis, Lozovoi, Alavi another story). Creating more charged defects (or introducing aliovalent dopants) affects Fermi energy. Need to Self-Consistently determine µ e!

16 Grand Canonical Minimisation Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina We can see that Concentrations sum to 1: i c i = 1. Total charge is Q = i c iq i. Total number of Mg atoms is i c i n Mg i. Total number of Ti atoms is i c i n Ti i. Obtain c i s that minimise G, subject to constraints: Overall charge neutrality Q = 0 Target level of Mg doping i c i n Mg i = [Mg] or Target level of Ti doping i c i ni Ti = [Ti] Transforms chemical potentials µ e, µ Mg, and µ Ti into Lagrange Multipliers.

17 Grand Canonical Minimisation Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Total Gibbs Free Energy G of system of N unit cells is { I } I G = N c i g i + k B T c i ln(c i /m i ) i=0 i=0 (5) c i : Concentration of cells containing defect of type i g i : Gibbs energy of cell containing defect of type i m i : Multiplicity of defect type i. Minimise G w.r.t. N and the c i s subject to constraints I + 1 equations, I + 1 unknowns

18 Cluster Mass Action Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina Can also obtain Law of Mass Action for formation of clusters, eg for we get Mg Al 1 + V O 2+ (Mg Al : V O ) 1+ (6) [Mg Al 1 ][V O 2+ ] [(Mg Al : V O ) 1+ ] = k 0e g f /kt (7) where k 0 is the ratio of multiplicities of each defect. Hence [(Mg Al : V O ) 1+ ] = 1 k 0 e g f /kt [Mg Al 1 ][V O 2+ ] (8)

19 Outline Introduction Self-Consistency 1 Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 2 Self-Consistency 3 Migration Barriers Calculating Effects of Aliovalent Doping 4

20 Methods Introduction Self-Consistency Calculate formation energies of lots of defects! Plane wave DFT with CASTEP. LDA (+PBE for comparison), USP s, 550eV PW cutoff. Full geometry relaxation of each structure. Finite Size extrapolation to infinite dilution limit ( atom cells). FLA approach to chemical potentials. No band-gap corrections. DFPT with harmonic approximation for g(t ) for each cell. For more details see N.D.M Hine, K. Frensch, W.M.C. Foulkes, and M.W. Finnis, Supercell size scaling of density functional theory formation energies of charged defects, Phys. Rev. B 79, (2009).

21 Formation Energies Introduction Self-Consistency First step is to calculated defect formation energies parameterised by µ e, µ α for all possible contributing species: G f i = G def i G perf X n α i i µ α + q i µ e α NB: Most papers stop here! But no use to experimentalists yet...

22 Cluster Binding Energies Self-Consistency Binding energies of several significant cluster species: Cluster Species g f (ev) (Mg Al : V O ) (Ti Al : V Al ) (Mg Al : Al 2+ i 3.67 (2Ti Al : V Al ) V 1 AlO 3.06 V AlO 1 may have a significant intrinsic population...

23 Self-Consistent Concentrations Self-Consistency Algorithm Specify q i, m i, n α i for each defect Input G f i (T ) for each defect Specify µ Al and µ O from external conditions, T, p O2. Provide initial guesses for µ e (Valence Band Maximum) and µ Mg (E T (Mg s )) Loop over µ e in the range E VBM to E CBM. Loop over µ Ti Calculate G f i and hence c i for each defect. Calculate Q = i c iq i, [Mg] = i c i n Mg i. Stop if Q = 0 and [Mg] = [Mg] targ. Record final c i s and µ e. Very easy to code and fast to implement

24 Self-Consistent Concentrations Self-Consistency Concentrations of substitutionals with increasing doping

25 Self-Consistent Concentrations Self-Consistency Concentrations of substitutionals, vacancies and clusters

26 Self-Consistent Concentrations Self-Consistency Concentrations of all significan defects, and Fermi level

27 Buffering Explained Introduction Self-Consistency Recall that [(Mg Al : V O ) 1+ ] = 1 k 0 e g f /kt [Mg Al 1 ][V O 2+ ] (9) Overall [Mg] Undoped [V 3 ] Al [V 2+ O ] const [V 3 ] Al u const [V 2+ O const [V 3 ] u + δ const [V 2+ [Mg1 ] [(Mg Al Al : V O ) 1+ ] ɛ F ]u near zero near zero stable Low Al ]u δ O rising [Mg] rising [Mg] stable Intermediate falling [Mg] 1 rising [Mg] rising [Mg] rising [Mg] 2 falling High const 0 const [V 2+ O ] f rising [Mg] rising [Mg] stable

28 Outline Introduction Migration Barriers Calculating Effects of Aliovalent Doping 1 Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 2 Self-Consistency 3 Migration Barriers Calculating Effects of Aliovalent Doping 4

29 DFT Migration Barriers Migration Barriers Calculating Effects of Aliovalent Doping Move atoms step by step from one possible site to an adjacent one Constrain atom to series of planes perpendicular to vector joining start and end points. Relax all atoms fully at each fraction along vector.

30 DFT Migration Barriers Migration Barriers Calculating Effects of Aliovalent Doping Migration barriers to site-to-site diffusion of oxygen vacancy Different paths show very different barriers. Lowest barriers 1eV only permits movement around smaller triangles of O 2 ions. Real barrier to 3D diffusion is 1.73eV.

31 DFT Migration Barriers Migration Barriers Calculating Effects of Aliovalent Doping Migration barriers to site-to-site diffusion of aluminium vacancy Revealed a whole new unexpected configuration (of notably lower energy) Split vacancy along c-axis.

32 Split Aluminium Vacancy Migration Barriers Calculating Effects of Aliovalent Doping Can be thought of as 2 vacancies and 1 interstitial...

33 Split Aluminium Vacancy Migration Migration Barriers Calculating Effects of Aliovalent Doping Moves in complex correlated motion of several atoms. Overall barrier is relatively low.

34 Migration Barriers Introduction Migration Barriers Calculating Effects of Aliovalent Doping of migration barriers for intrinsic defect species: Defect Species E mig (ev) V O V Al Al i O i V AlO Reasonably in line with expectations of ceramicists (much lower than previous estimates which presumably did not find the fully relaxed path)

35 Calculating Migration Barriers Calculating Effects of Aliovalent Doping Diffusion coefficients D i given by Arrhenius equation: D i = D 0 e E a i /k B T, (10) where the activation energy Ei a is the sum of the migration energy and the formation energy. Pre-exponential factor D 0 f α 2 ν, f : correlation factor (calculate from structure) α: jump distance (estimate from bond lengths) ν: attempt frequency (estimate from phonon frequencies)

36 Migration Barriers Calculating Effects of Aliovalent Doping Reliable Experimental Diffusion Measurements Loop annealing Data (Heuer), Al Tracer data (Fielitz)

37 Oxygen Migration Barriers Calculating Effects of Aliovalent Doping Calculated vs Experimental diffusion coefficients for V O.

38 Aluminium Migration Barriers Calculating Effects of Aliovalent Doping Calculated vs Experimental diffusion coefficients for V Al and Al i.

39 Conundrum Solved? Migration Barriers Calculating Effects of Aliovalent Doping Buffering effect observed under aliovalent doping in oxygen and aluminium diffusion coefficient explained in terms of movement of Fermi level with changing ratios of concentrations of substitutionals relative to intrinsic defects and clusters. Predicted and observed activation energies reconciled: good quantitative predictions of diffusion coefficients made possible in ceramics.

40 Outline Introduction 1 Introduction Intrinsic defects in Alumina Doping and Clusters Diffusion in Alumina 2 Self-Consistency 3 Migration Barriers Calculating Effects of Aliovalent Doping 4

41 Introduction A new approach to Self-Consistent determination of defect concentrations and diffusion coefficients. Explains the long-standing mysteries of diffusion in Alumina. Ab Initio calculations of diffusion coefficients agree well with experiments (to 1-2 orders of magnitude).

42 Outlook Introduction Other common dopants? Cr, Fe? (requires some treatment of correlated d electrons) Ever larger clusters & larger supercells (towards ONETEP regime) Solubilities work in progress (KF) Did we get the right structure for all defects? (esp O i ) Can we do Grain Boundaries rich, interesting, unexplained physics Opens up diffusion in ceramics to the same kind of engineering of desired properties possible with doping in semiconductors.